Analysis of the impact of parameters values on the...

Analysis of the impact of parameters values

on the Genetic Algorithm for TSP

Avni Rexhepi1, Adnan Maxhuni2, Agni Dika3

1,2,3 Faculty of Electrical and Computer Engineering, University of Pristina, Kosovo

Abstract

Genetic algorithms (GAs) are multi-dimensional and stochastic

search methods, involving complex interactions among their

parameters. Researchers have been trying to understand the

mechanics of GA parameter interactions by using various

techniques. It still remains an open question for practitioners as

to what values of GA parameters (such as population size, choice

of GA operators, operator probabilities, and others) to use in an

arbitrary problem.

Genetic algorithm (GA) parameters are explored to minimize the

time needed to find a solution. The basic GA code stays the same

throughout the entire system. Variable parameters include

mutation rate, crossover rate, crossover operator, number of

generations, population size, etc. When an optimization problem

is encoded using genetic algorithms, one must address issues of

population size, crossover and mutation operators and

probabilities, stopping criteria, selection operator and pressure,

and fitness function to be used in order to solve the problem.

This paper tests a relationship between (1) size of initial

population, (2) mutation probability, and (3) number of

generations in runs of Genetic Algorithm for solving the TSP.

TSP is an NP hard problem, so using Genetic Algorithm we can

find a solution on reasonable amount of time. In this paper we

describe the results of the solution for the Traveling Salesman

Problem (TSP) for Kosovo municipalities, using the genetic

algorithm, with different settings for the parameters of the

Genetic Algorithm [12].

Keywords: Genetic Algorithms, TSP, Parameter Selection,

Initial Population, Crossover, Mutation

1. Introduction

TSP is a problem, where traveling salesman wants to visit

each of a set of cities exactly once, starting from

hometown and returning to his hometown. His problem is

to find the shortest route for such a trip. TSP has a model

character in many branches of Mathematics, Computer

Science, Operations Research, etc. Linear programing,

heuristics and branch and bound which are main

components for the most successful approaches to hard

combinatorial optimization problems, were first formulated

for the TSP and used to solve practical problem instances

in 1954 by Dantzig, Fulkerson and Johnson.

When the theory of NP-completeness developed, the TSP

was one of the first problems to be proven NP-hard by

Karp in 1972. New algorithmic techniques have first been

developed for or at least have been applied to the TSP to

show their effectiveness. Such examples are branch and

bound, Lagrangean relaxation, Lin-Kernighan type

methods, simulated annealing, etc. [3].

Representation model is: Let Kn=(Vn, En) be the complete

undirected graph with n=|Vn| nodes and m=|En|=

2

n

edges. An edge e with endpoints i and j is also denoted by

ij, or by (i,j). We denote by REn

the space of real vectors

whose components are indexed by the elements of En. The

component of any vector nz ER indexed by the edge

e=ij is denoted by ze, zij, or z(i,j).

Given an objective function nc ER , that associates a

“length” ce with every edge e of Kn, the symmetric

traveling salesman problem consists of finding a

Hamiltonian cycle such that its c-length (the sum of the

lengths of its edges) is as small as possible.

Of special interest are the Euclidean instances of the

traveling salesman problem. In these instances the nodes

defining the problem correspond to points in the two-

dimensional plane and the distance between two nodes is

the Euclidean distance between their corresponding points.

IJCSI International Journal of Computer Science Issues, Vol. 10, Issue 1, No 3, January 2013 ISSN (Print): 1694-0784 | ISSN (Online): 1694-0814 www.IJCSI.org 158

Copyright (c) 2013 International Journal of Computer Science Issues. All Rights Reserved.

More generally, instances that satisfy the triangle

inequality, i.e., ikjkij ccc for all the three distinct i,j

and k, are of particular interest.

For our case, we consider the locations of the

cities/municipalities in Kosovo map as nodes of the graph.

For to do this, we take their geographic coordinates and

then based on that, we calculate their position in our map

scaled to a smaller size, for to calculate the real positions

and distances, by using the real life values for distances in

kilometers between the cities.

2. Genetic Algorithms

Genetic Algorithms (GA) [1,2] are computer algorithms

that search for good solutions to a problem within a large

number of possible solutions. They were proposed and

developed in the 1960s by John Holland, his students, and

his colleagues at the University of Michigan. These

computational paradigms were inspired by the mechanics

of natural evolution, including survival of the fittest,

reproduction, and mutation. These mechanics are well

suited to resolve a variety of practical problems, including

computational problems, in many fields. Some applications

of GAs are optimization, automatic programming, machine

learning, economics, immune systems, population genetic,

and social system.

GAs have been successfully applied to many problems of

business, engineering, and science. Because of their

operational simplicity and wide applicability, GAs play an

important role in computational optimization and

operations research [6].

The genetic algorithm transforms a population (set) of

individual objects, each with an associated fitness value,

into a new generation of the population using the

Darwinian principle of reproduction and survival of the

fittest and analogs of naturally occurring genetic operations

such as crossover (sexual recombination) and mutation.

Each individual in the population represents a possible

solution to a given problem. The genetic algorithm

attempts to find a very good (or best) solution to the

problem by genetically breeding the population of

individuals over a series of generations.

2.1 Basic elements of GAs

Most GAs methods are based on the following elements:

populations of chromosomes, selection according to

fitness, crossover to produce new offspring, and random

mutation of new offspring. The chromosomes in GAs

represent the space of candidate solutions. Possible

chromosomes encodings are binary, permutation, value,

and tree encodings. GAs require a fitness function which

allocates a score to each chromosome in the current

population. Thus, it can calculate how well the solutions

are coded and how well they solve the problem [2].

The selection process is based on fitness. Chromosomes

that are evaluated with higher values (fitter) will most

likely be selected to reproduce, whereas, those with low

values will be discarded. The fittest chromosomes may be

selected several times, however, the number of

chromosomes selected to reproduce is equal to the

population size, therefore, keeping the size constant for

every generation. This phase has an element of randomness

just like the survival of organisms in nature. The most used

selection methods, are roulette-wheel, rank selection,

steady-state selection, and some others. Moreover, to

increase the performance of GAs, the selection methods

are enhanced by elitism. Elitism is a method, which first

copies a few of the top scored chromosomes to the new

population and then continues generating the rest of the

population. Thus, it prevents losing the few best found

solutions.

Crossover is the process of combining the bits of one

chromosome with those of another to create an offspring

for the next generation that inherits traits of both parents.

Mutation is performed after crossover to prevent falling all

solutions in the population into a local optimum of solved

problem.

So, general outline of basic GA is:

1. Start: Randomly generate a population of N

chromosomes.

2. Fitness: Calculate the fitness of all chromosomes.

3. Create a new population:

a. Selection: According to the selection method select 2

chromosomes from the population.

b. Crossover: Perform crossover on the 2 chromosomes

selected.

c. Mutation: Perform mutation on the chromosomes

obtained.

4. Replace: Replace the current population with the new

population.

5. Test: Test whether the end condition is satisfied. If so,

stop. If not, return the best solution in current population

and go to Step 2.

Each iteration of this process is called generation.

The genetic algorithm object determines which individuals

should survive, which should reproduce, and which should

die. It also records statistics and decides how long the

evolution should continue. A typical genetic algorithm will

run forever, so we must build functions for specifying

when the algorithm should terminate. These include

terminate-upon generation, in which you specify a certain

number of generations for which the algorithm should run,

and terminate-upon-convergence, in which you specify a

value to which the best-of-generation score should

converge. One can customize the termination function to

use own stopping criterion and must tell the algorithm



when to stop. Often the number-of generations is used as a

stopping measure, but you can use goodness-of-best-

solution, convergence-of-population, or any problem-

specific criterion if you prefer.

There are some flavors of genetic algorithms. For example,

the first is the standard 'simple genetic algorithm' described

by Goldberg in his book [2]. This algorithm uses non-

overlapping populations and optional elitism. Each

generation the algorithm creates an entirely new population

of individuals. The second is a 'steady-state genetic

algorithm' that uses overlapping populations. In this

variation, you can specify how much of the population

should be replaced in each generation. The third variation

is the 'incremental genetic algorithm', in which each

generation consists of only one or two children. The

incremental genetic algorithms allow custom replacement

methods to define how the new generation should be

integrated into the population. So, for example, a newly

generated child could replace its parent, replace a random

individual in the population, or replace an individual that is

most like it. The fourth type is the 'deme' genetic

algorithm. This algorithm evolves multiple populations in

parallel using a steady-state algorithm. Each generation the

algorithm migrates some of the individuals from each

population to one of the other populations.

The base genetic algorithm class contains operators and

data common to most genetic algorithms.

The genetic algorithm contains the statistics, replacement

strategy, and parameters for running the algorithm. The

population object, a container for genomes, also contains

some statistics as well as selection and scaling operators.

The number of function evaluations is a good way to

compare different genetic algorithms with various other

search methods[3]. The basic algorithm is as follows:

Figure 1 – Genetic Algorithm

2.2 TSP

In order to calculate the shortest traveling distance from an

initial city, by visiting each one only once and returning to

the initial one, we consider the locations as nodes of the

graph, in the graph model. In the meantime, the distances

between the cities are edges of the graph.

For length of the edges we take inter-city distances in

kilometers. We have created a matrix of distances between

all the municipalities, where the matrix elements cij are

elements of the square symmetrical matrix, since distance

ij is equal to the distance in the other side ji. The diagonal

of the matrix will contain zero values, since diagonal

elements of the matrix cij, where i=j, will represent the

distance of the city to itself, so in fact it will be the

traveling distance of zero kilometers, therefore these

elements will be equal to cij=0.

By using complete graph in the definition of the TSP, the

existence of a feasible solution is guaranteed, while for

general graphs deciding the existence of a Hamiltonian

cycle is an NP-complete problem. The number of

Hamiltonian cycles in Kn, i.e. the size of the set of feasible

solutions of the TSP is (n-1)!/2.

The algorithmic treatment of the TSP ensures an

approximation algorithm that cannot guarantee to find the

optimum, but which is the only available technique to find

a good solution to a large problem instances. To assess the

quality of a solution, one has to be able to compute a lower

bound on the value of the shortest Hamiltonian cycle.

We have built an application in C#, with an image with the

small-scaled size of the Kosovo map, with depicted

municipality boundaries and locations (Figure 2).

It is possible to select a particular city by clicking on the

map and we have also added buttons that make it possible

to create locations for the biggest cities and locations for

the all municipalities.

Figure 2 – Screenshot of the application



We used the geographical coordinates of the cities, to

calculate their positions. By running the application, we

can calculate the shortest traveling distance between the

selected cities, by finding a solution of the TSP using a

genetic algorithm.

User can set up the values for initial population, maximal

number of generations, size of the group, percentage of

mutations and number of close cities/locations (used by the

algorithm, while finding closest locations).

3. Simulation and Results

We analyzed the results of different cases with different

values for the parameters of initial population, probability

of mutation and number of generations.

Firstly, we have set the value of the maximal number of

generations to 10,000 and we calculated the fitness for the

cases where the size of the initial population is: 1000, 5000

and 10000 and for each of these cases, we calculated the

results for the mutation rate of: 1%, 3%, 5% and 10% (as

in Figures: 3, 4 and 5).

Then, we compared the results for each value of the

mutation rate (1%, 3%, 5% and 10%), with different values

of the initial population parameter values (1000, 5000 and

10000, as in Figures: 6, 7, 8 and 9).

We repeated this for the values of maximal number of

generations of 50,000 and 100,000, too (see Appendix).

In each figure, the y-axis is the value of the fitness and the

x-axis is the maximal number of generations, which serves

as the interruption parameter for the genetic algorithm.

Figure 3 – Initial population size: 1000; pm=1%, 3%, 5% and 10%.



Now we present the results for the cases when, for some

particular probability of mutation, we take different values

for the parameter of the Initial Population. In Figure 6 we

show the changes in fitness for different values of the

number of initial population, having the same probability

of mutation (pm=1%). Similarly, for other pm values (3%,

5% and 10%), in Figure 7, 8 and 9.

Figure 6 – pm=1%,; Initial population sizes: 1000, 5000 and 10000.



Figure 7 - pm=3%,; Initial population sizes: 1000, 5000 and 10000



4. Conclusions

Since the use of correct population size is a crucial factor

for successful GA applications, more efforts need to be

spent in finding correct population sizing estimates for

particular problems. Our results show that there is a slight

or no difference in fitness results for the case of TSP with

GA, for different values of the initial population size, with

all tested values of mutation parameter. So increasing the

mutation rate doesn’t contribute to a better results.

There is no use of increasing the value of the probability of

mutation, since it doesn’t pay-off as it increases the

execution time/processor time and doesn’t contribute to a

much better fitness result.

Considering the effect of changing the initial population

size for fixed mutation rate, we see that the effect of the

mutation is important for small initial populations, since it

contributes to promoting new solutions in the solution

space.

For each percentage of the mutation, results show that the

bigger the value of the initial population, the less is the

fitness value improved.

The effect of mutation is especially noticed in the first

generations of the cases with small value of the intimal

population (when there is no enough number of good

solutions). Otherwise, when the initial population is of

higher value, it means that there is a higher diversity of

solutions and so the mutation doesn’t bring much new and

better solutions.

When the population is large, the diversity in the initial

random population is large and the best solution in the

population is expected to be close to the optimal solution.

Appendix

Here we will present the results for the cases when the

value of the parameter of maximal number of generations

is: 50,000 and 100,000.

For each value of the number of generations, we present

the results for cases with values of the initial population

(1000, 5000 and 10000) for different values of the

probability of mutation (1%, 3%, 5% and 10%) and then

cases with same probabilities of mutation for different

values of initial population (1000, 5000 and 10000) .



References

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Avni Rexhepi, MSc. (DB: 12/12/1969). Master of Science in computer science (2004) - University of Prishtina. Teaching and research Assistant in the University of Prishtina, since 1996.

Adnan Maxhuni, MSc. (DB: 31/12/1967). Master of Science in computer science (2005) - University of Prishtina. Teaching and research Assistant in the University of Prishtina, since 1993.

Agni Dika, Prof. Dr. (DB: 21/02/1950). PhD in computer sciences (1989) - University of Zagreb, Croatia. Ordinary Professor at the Faculty of Electrical and Computer Engineering, in the University of Prishtina.



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